Why does a neuron in a multi-layer network need several input connections?
There's a few reasons I can think of, though I have not read an explicit description of why it is done this way. It's likely that people just started doing it this way because it's most logical, and people who have attempted to try your method of having reduced connections have seen a performance hit and so no change was made.
The first reason is that if you allow all nodes from one layer to connect to all others in the next, the network will optimise unnecessary connections out. Essentially, the weighting of these connections will become 0. This, however, does not mean you can trim these connections, as ignoring them in this local minima might be optimal, but later it might be really important these connections remain. As such, you can never truly know if a connection between one layer and the next is necessary, so it's just better to leave it in case it helps improve network performance.
The second reason is it's just simpler mathematically. Networks are implemented specifically so it's very easy to apply a series of matrix calculations to perform all computations. Trimming connections means either:

A matrix must contain 0 values, wasting computation time
A custom script must be written to calculate this networks structure, which in the real world can take a very long time as it must be implemented using something like CUDA (on a GPU level, making it very complicated)

Overall, it's just a lot simpler to have all nodes connected between layers, rather than on connection per node.